Tolman IV fluid sphere in bigravity Ksh. Newton Singh • Susmita Sarkar • Farook Rahaman Abstract We present Tolman IV spacetime represent- 1 Introduction ing compact fluid sphere in bigravity. Here we have ex- plored the effect of scale parameter k in the local matter A long years back, N. Rosen (Rosen 1973, 1974, 1975, distribution of compact stars. We have model for three 1978) proposed a new modified theory to Einstein’s well-known compact stars and it shows that for lower general theory of relativity (GTR) involving a back- values of k leads to stiffer EoS. This claim is also sup- ground metric γµν in addition to the usual physical ported by the graphical analysis. It can be observed metric gµν , which is known as the “bimetric general that the sound speed and the adiabatic index are more theory of relativity (BGTR or bigravity)”. The back- for lower values of k. It is also seen that all the solu- ground metric γµν is governed by a parameter k. In tions of Einstein’s field equations are still satisfying the BGTR, the field equations are similar of Einstein’s field equations in the presence of a background metric GTR however, with the ordinary derivatives of phys- γµν . However, the density and pressure does modified ical metrics are replaced by the covariant derivatives by extra term from the constant curvature background, with respect to the background metric. Both gµν and thus affecting the EoS. One can also think that the pa- γµν are present in the field equations but only gµν in- rameter α 1/k2 as coupling constant between the g ≡ µν teracts with matter. In the year 1978, Rosen (1978) and γµν and consequently more the coupling stiffer is initially written the static field equations in the form the EoS. As k , the background de-Sitter space- → ∞ of Einstein’s field equations with an additional bimet- time reduces to Minkowski’s spacetime and the coupling ric term. In the year 1985, Harpaz & Rosen (1985) vanishes. The solution satisfy the causality condition, solved the field equations, which are obtained by us- all the energy conditions and equilibrium under gravity ing the procedure of Rosen (1980) corresponding to and hydrostatic forces. The stability of the local stellar de-Sitter universe of constant curvature, to obtain a structure is enhanced by reducing the scalar curvature arXiv:2006.07135v1 [gr-qc] 10 Jun 2020 model of compact star. They also proved that for of the background spacetime. an ordinary star, the results obtained from BGTR Keywords Bigravity ; Tolman IV ; Compact star ; has no difference from the result obtained from Ein- Stability stein’s GTR. However, for a collapsed star model the two theories yield different results. The BGTR be- comes a famous among the researcher within few years Ksh. Newton Singh of the proposal and various aspects were investigated Department of Physics, National Defence Academy, Khadak- wasla, Pune, Maharashtra-411023, India (Khadekar & Karade 1989; Reddy & Venkateswarlu email: [email protected] 1989; Goldman & Rosen 1977; Akrami et al. 2015). Susmita Sarkar It is investigated that the bimetric gravity has a sub- et al. Department of Mathematics, Jadavpur University, Kolkata, In- tle relationship with massive gravity (Baccetti dia. 2012). Studies of a finite size charge and the static email:[email protected] spherically symmetric field of an electric charge were Farook Rahaman done in the framework of BGTR in Rotbart (1979) Department of Mathematics, Jadavpur University, Kolkata, In- and Rosen & Rotbart (1979), respectively. In the dia. year 1981, Falik & Rosen (1981) have investigated a email:[email protected] 2 charged point particle and then Rosen, himself dis- been renewed interest in bigravity due to their accel- cussed a classical model for elementary particles in erating cosmological solutions (Damour et al. 2002). BGTR (Rosen 1989). Hassan & Rosen (2012) showed that the introduction In the context of the Hassan-Rosen theory, spheri- of a kinetic term for the background metric in the ghost- cally symmetric systems were first studied by Comelli et al. free massive gravity leads to a bigravity theory which (2012), where in particular, the perturbative solutions is also ghost free. A details study of cosmological solu- to the equations of motion was published. In pres- tions in bigravity can be found in refs. Volkov (2012); ence of bimetric gravity, Volkov (2012) performed Strauss et al. (2012); Comelli et al. (2012). an extensive numerical study, and gave conditions for In present paper, we have developed a model of com- the existence of asymptotically flat black hole solu- pact star in bimetric gravity where the metric potentials tions. Star solutions and the so-called Vainshtein are chosen as Tolman IV metric potential. In one of our mechanism was studied by Babichev & Crisostomi previous paper, we have investigated a new model of (2013). Solutions for charged black holes and for ro- anisotropic compact star in (3+1)-dimensional space- tating black holes can be found in Brito et al. (2013), time. The model was obtained in the background of Babichev & Fabbri (2014), respectively. In massive bi- Tolman IV grr metric potential as input and the field gravity, a general review of black holes was proposed by equatios were solved by assuming a suitable expression Babichev & Brito (2222). Enander & M¨ortsella pro- for radial pressure pr (Bhar et al. 2016). The present posed the phenomenology of stars and galaxies in mas- paper is organized as follows: In sect. 2, field equations sive bigravity by applying a parameter conditions for in bigravity is developed. In sect. 3 and 4, the model the existence of viable star solutions when the radius of parameters are obtained by solving the field equations the star is much smaller than the Compton wavelength and physical properties are discussed, respectively. In of the graviton. sect. 5, we match our interior spacetime to the exte- Avakian et al. (1991) have studied on superdense rior Schwarzschild line element. Final two sections are celestial objects in generalized bimetric gravity with devoted on a brief discussion. a variable gravitational constant. In the framework of bimetric gravity, the solution for spherically sym- 2 Field equations in bimetric gravity metric self-gravitating anisotropic matter distribution was investigated by Khadekar & Kandalkar (2004) The bigravity modifies the Einstein’s theory by ac- and that solution agrees with the Einstein’s GRT counting a background space-time. Therefore, one can for a physical system compared to the solar system defined the two symmetric tensor g and γ associ- size of universe. A charged fluid was also studied µν µν ated with the two spacetime and are defined as in BGTR (Kandalkar & Gawande 2010). Recently, et al. 2 µ ν 2 µ ν Grigorian (2018) have shown that the masses ds− = gµν dx dx , dsb = γµν dx dx . (1) for superdense compact objects in bimetric theory can be essentially larger compared to the objects in GTR These two metric tensor are having non-vanishing de- depending on the value of the bimetric parameter. terminants. Adopting the notation used by Rosen (1940), λ is the Chistoffel symbol in g different- It is well known that, in the presence of an extra µ ν − iation and Γλ in γ differentiation. The relationship spin-2 field, bimetric theory describes gravitational in- µν − teractions. Recently, Hassan et al. (Hassan & Rosen between these two symbols are given as 2012,a) have proposed a particular bimetric theory (or λ λ λ bigravity) to avoid the ghost instability. This theory µ ν =Γµν + ∆µν , (2) describes a nonlinear interactions of the gravitational n o λ metric with an additional spin-2 field. By assuming where, ∆µν is found to be related to gµν as the Planck mass Mf of the second metric, the ghost 1 ∆λ = gλσ g + g g . (3) instability can be avoid by pushing back to early un- µν 2 µσ,ν νσ,µ − µν,σ observably times. This limit the uses an effective cos- mological constant in general relativity (Akrami et al. Now the Riemann tensor related to the two metric func- 2015). tions can be written as It is quite familiar that non-linear bimetric theories λ = λ ∆λ + ∆λ + ∆λ ∆α ∆λ ∆α of gravity suffer from the same Boulware-Deser ghost Rµνσ Pµνσ − µν,σ µσ,ν αν µσ − ασ µν = ∆λ + ∆λ + ∆λ ∆α ∆λ ∆α . (4) instability (Boulware & Deser 1972). To describe the − µν,σ µσ,ν αν µσ − ασ µν interaction of gravity with a massive spin-2 meson, One can easily see that the tensor ∆λ is related to such theories were introduced. Recently, there has µν the normal Riemann curvature tensor. The curvature 3 1.00 450 0.95 0.90 -λ 400 e 3 0.85 fm / 0.80 MeV 350 k = 100 Blue 0.75 ρ = Metric Potentials k 150 Red 0.70 k = 200 Black e ν 300 0.65 2 4 6 8 0 2 4 6 8 r km r km Fig. 1 Varaition of metric functions with respect to the Fig. 2 Variation of density with respect to the radial co- radial coordinate r for LMC X-4 (M = 1.04M⊙, R = 8.3km ordinate r for LMC X-4 (M = 1.04M⊙, R = 8.3km and and b = 0.001896, B = 0.788065) with (a = 0.00266, k = b = 0.001896, B = 0.788065) with (a = 0.00266, k = 100), 100), (a = 0.0029, k = 150) and (a = 0.003, k = 200). (a = 0.0029, k = 150) and (a = 0.003, k = 200). tensor of the constant curvature spacetime can Following Falik & Rosen (1980) we can write the Pλµνσ be written as field equations as 1 ′ −ν −λ λ 1 1 3e λµνσ = γµν γλσ γµσγλν .